B X q l + l (q = d,s) and Determination of V td

Transcription

1 YUMS SNUTP HUPD-9710 DPN B X q l + l (q = d,s) and Determination of V td V ts arxiv:hep-ph/ v2 12 Aug 1997 C. S. Kim Department of Physics, Yonsei University Seoul , Korea T. Morozumi Department of Physics, Hiroshima University Kagamiyama, Higashi Hiroshima, Japan 739 A. I. Sanda Department of Physics, Nagoya University Chikusa-ku, Nagoya, Japan Abstract We propose a new method to extract V td V ts from the ratio of the decay distributions dr ds db ds [B X dl + l ]/ db ds [B X sl + l ]. This ratio depends only on the KM ratio V td V ts with 15% theoretical uncertainties, if dilepton invariant mass-squared s is away from the peaks of the possible resonance states, J/ψ, ψ, and etc. We also give a detailed analytical and numerical analysis on db ds [B X ql + l ](q = d,s) and dr ds. address: kim@cskim.yonsei.ac.kr, cskim@kekvax.kek.jp address: morozumi@theo.phy.sci.hiroshima-u.ac.jp address: sanda@eken.phys.nagoya-u.ac.jp

2 1 Introduction The determination of the elements of KM matrix is one of the most important issues in the quark flavor physics. Moreover, the element V td (or V ub ) is especially important to the Standard Model description of CP-violation. If it were zero, there would be no CP-violation from the KM matrix elements (i.e. within the Standard Model), and we have to seek for other source of CP violation in K L ππ. Here we study the ratio V td V ts. In the Standard Model with the unitarity of KM matrix, V ts is approximated by V cb, which is directly measured by semileptonic B decays. There are already several ways to determine V td available in the literature: V td can be indirectly extracted through B d B d mixing. However, in B d B d mixing the large uncertainty of the hadronic matrix elements prevents us to extract the element of KM with good accuracy. A better extraction of V td V ts can be made if B s B s is measured as well, because the ratio f 2 B B Bd /f 2 B can be much better determined. d Bs Bs The determination of V td V ts from the ratios of rates of several hadronic two-body B decays, such as Γ(B 0 K 0 K 0 )/Γ(B 0 φk 0 ), Γ(B 0 K 0 K 0 )/Γ(B 0 φk 0 ), Γ(B + K 0 K + )/Γ(B + φk + ), and Γ(B + K 0 K + )/Γ(B + φk + ) has been also proposed in Ref. [1]. V td can be determined from K + π + ν ν decay within 15% theoretical uncertainties with the branching fraction B [2]. Why do we discuss yet another method to determine V td? The main reason we are interested in B physics is that this area is very likely to yield information about new physics beyond the Standard Model. We expect that new physics will influence experimentally measureable quantities in different ways. For example, most of us expect that B = 2 transition is more sensitive to new physics than the decay rates. New physics may couple differently to K mesons compared to B mesons. Therefore, it is essential to determine the KM matrix elements in as many different methods as possible. In this paper, we propose another method to determine V td V ts precisely from the decay distributions db [B X ds sl + l ] and db [B X ds dl + l ], where s is invariant mass- 2

3 squared of final lepton pair, l + l. In the decays of B X q l + l (q = d,s), the short distance (SD) contribution comes from the top quark loop diagrams and the long distance (LD) contribution comes from the decay chains due to intermediate charmonium states. Therefore, the former (SD) amplitude is proportional to V tq V tb, and the latter (LD) proportional to V cq V cb. If the invariant mass-squared of l + l is away from the peaks of the charmoniuum resonances (J/ψ and ψ ), the SD contribution is dominant, while on the peaks of the resonances the LD contribution is dominant, and therefore we expect that in the SU(3) symmetry limit the ratio becomes dr ds db ds (B X dl + l )/ db ds (B X sl + l ) 1 dr(s) λ 2 ds (1 ρ) 2 +η 2 (s 1GeV 2 ), 1 (s m J/ψ 2 ), (1) where λ is sinθ c, θ c = Cabibbo angle, and for ρ, η, see Eq. (8). In the intermediate region, there is a characteristic interference beteen the LD contribution and the SD contribution, which requires the detailed study of the distributions. By focusing on the region, 1(GeV 2 ) s < m 2 J/ψ, we can study V td V ts from the experimental ratio of the distributions dr ds. We stress the advantages to use the inclusive semileptonic decays: If dilepton invariant mass-squared s is away from the peaks of the possible intermediate states, J/ψ, ψ, ρ, ω, and etc., the short distance contribution due to top quark loop is dominant. Therefore, we may extract very precisely the combination of V td V ts from the decay distributions within the range 1(GeV 2 ) s < m J/Ψ 2 without anytheoretical uncertainties fromtheldhadronicmatrices, unknown KM matrix elements, and etc. Inclusive decay distributions are theoretically well predicted by heavy quark mass expansion away from the phase space boundary. The leading order of the expansion agrees with the parton model result. Furthermore, non-perturbative 1/m b 2 power corrections of QCD can be easily incorporated. This paper is organized as follows. In Section 2, we present the analytic formulae for db ds [B X sl + l ] and db ds [B X dl + l ]. We also show in detail the decay distri- 3

4 bution db ds [B X sl + l ] including the uncertainties coming from top quark mass m t and the scale of renormalization group µ. In Section 3, after brief discussion on the limit of KM elements coming from B d B d mixing and B X u l ν, we study the ratio of the decay distributions, and show how we may extract V td V ts. 2 Effective Hamiltonian for b ql + l and their differential decay rates. In this Section, the differential decay rates for B X q l + l (q = d,s) are shown including both LD and SD effects as well as Λ QCD 2 /m b 2 power corrections. (See [3] for b dl + l, [4] for b sl + l, and [5], [6] for B X s l + l including the 1/m b 2 power corrections.) We also show in detail how the differential decay rate varies by changing the input parameters m t, and the renormalization scale µ. In Table 1, we summarize all the values of the input parameters used in our numerical calculations of decay rates. We use the central values for those input parameters, unless otherwise specified. The effective Hamiltonian for b ql + l (q = d,s) is given as H eff = 4G [ 10 ] F VtqV tb C i O i 2 i=1 + 4G F Vuq V ub[c 1 (O u 1 O 1 )+C 2 (O u 2 O 2 )], (2) 2 where V ij are the KM matrix elements. The operators are given as O 1 = ( q Lα γ µ b Lα )( c Lβ γ µ c Lβ ), O 2 = ( q Lα γ µ b Lβ )( c Lβ γ µ c Lα ), O 3 = ( q Lα γ µ b Lα ) ( q Lβγ µ q Lβ ), O 4 = ( q Lα γ µ b Lβ ) O 5 = ( q Lα γ µ b Lα ) O 6 = ( q Lα γ µ b Lβ ) O 7 = O 8 = q =u,d,s,c,b q =u,d,s,c,b q =u,d,s,c,b q =u,d,s,c,b ( q Lβγ µ q Lα), ( q Rβγ µ q Rβ ), ( q Rβγ µ q Rα ), e 16π 2 q ασ µν (m b R+m q L)b α F µν, g 16π 2 q αtαβσ a µν (m b R+m q L)b β G aµν, 4

5 O 1 u = ( q Lα γ µ b Lα )(ū Lβ γ µ u Lβ ), O 2 u = ( q Lα γ µ b Lβ )(ū Lβ γ µ u Lα ), O 9 = e2 16π 2 q αγ µ Lb α lγµ l, O 10 = e2 16π 2 q αγ µ Lb α lγµ γ 5 l, (3) where L and R denote chiral projections, L(R) = 1/2(1 γ 5 ). We use the Wilson coefficients given in the literature (see, for example, [7]). With the effective Hamiltonian in Eq. (2), the matrix element for the decays b ql + l (q = d,s) can be written as M(b ql + l ) = G Fα 2π V tq V tb where C eff 9q is given by C eff 9q (ŝ) C 9 { [( C eff ) 9q C 10 ( qγ µ Lb) ( lγ µ Ll ) ( + C eff ) 9q +C 10 ( qγ µ Lb) ( lγ µ Rl ) 2C eff 7 1+ α } s(µ) π ω(ŝ) ( q ν ) qiσ µν q 2(m ql+m b R)b ( lγ µ l )], (4) +Y SD q (ŝ)+y LD q (ŝ), (5) where q ν is the four momentum of l + l, s = q 2, and ŝ = s/m b 2. The function Y SD q (ŝ) is the one-loop matrix element of O 9, and Y LD q (ŝ) is the LD contributions due to the vector mesons mesons J/ψ, ψ and higher resonances. The function ω(ŝ) represents the O(α s ) correction from the one-gluon exchange in the matrix element of O 9, and is given in our Appendix. The two functions Y SD q and Y LD q are written as Y SD q (ŝ) = g(ˆm c,ŝ)(3c 1 +C 2 +3C 3 +C 4 +3C 5 +C 6 ) 1 2 g(1,ŝ)(4c 3 +4C 4 +3C 5 +C 6 ) 1 2 g(0,ŝ)(c 3 +3C 4 ) (3C 3 +C 4 +3C 5 +C 6 ) V uqv ub (3C VtqV 1 +C 2 )(g(0,ŝ) g(ˆm c,ŝ)), (6) tb Y q LD (ŝ) = 3 ( α 2κ V cq V cb C (0) V uq V ) ub (3C VtqV tb VtqV 3 +C 4 +3C 5 +C 6 ) tb V i =ψ(1s),...,ψ(6s) πγ(v i l + l )M Vi M Vi 2 ŝm b2 im Vi Γ Vi, (7) 5

6 where the function g( ˆm c = mc m b,ŝ), g(1,ŝ), and g(0,ŝ) represent c quark, b quark and u, d, s quark loop contributions, respectively. The two functions g(z, ŝ) and g(0, ŝ) are given [7] in our Appendix. In Eq. (7), C (0) 3C 1 +C 2 +3C 3 +C 4 +3C 5 +C 6, and the first two terms are dominant, as can be seen from the Table 2. It is convenient to write the relevant combinations of KM in terms of the Wolfenstein parametrization. In the following, in addition to A andλ, we choose (1 ρ) 2 +η 2 and φ 1 = arg( V cdvcb ) as independent variables, then we have V td Vtb Vts V tb λ 2 A, Vcs V cb V tsv tb 1, Vus V ub O(λ 2 ), VtsV tb V VcdV cb td V tb λ 3 A(1 ρ+iη), 1 Vtd V tb 1 ρ+iη, = exp( iφ 1) (1 ρ) 2 +η 2, V ud V ub V td V tb ρ iη 1 ρ+iη, = 1+ exp( iφ 1) (1 ρ) 2 +η 2. (8) In the SD contribution of B X s l + l, the u-quark loop contribution is neglected due to the smallness of the combination V us V ub compared with V cs V cb V ts V tb, while in B X d l + l the term which is proportional to V ud V ub is maintained. In the LD contribution, there is a contribution coming from the gluonic penguin amplitude which is proportional to V ud V ub V td V tb (3C 3+C 4 +3C 5 +C 6 ). This contribution is neglected because of the smallness of the Wilson coefficient (3C 3 +C 4 +3C 5 +C 6 ) compared with C (0). We adopt κ = 2.3 [8] to reproduce the rate of the decay chain B X s J/ψ X s l + l. Note that the data determines only the combination, κc (0) = By combining both SD and LD contributions as well as non-perturbative 1/m b 2 power corrections, the differential decay rate for B X q l + l (q = d,s) becomes db dŝ = 2 V V cb 2 B 0 tq V tb 2 {[ 2 3û(ŝ, ˆm q)((1 ˆm 2 q )2 +ŝ(1+ ˆm 2 q ) 2ŝ2 ) (1 4ˆm2 q +6ˆm 4 q 4ˆm 6 q + ˆm 8 q ŝ+ ˆm 2 qŝ+ ˆm 4 qŝ ˆm 6 qŝ 6

7 ˆλ 3ŝ 2 2ˆm 2 qŝ 2 3ˆm 4 qŝ 2 +5ŝ 3 +5ˆm 2 qŝ 3 2ŝ 4 1 ) û(ŝ, ˆm q ) + ( 1 8ˆm 2 q +18ˆm 4 q 16ˆm 6 q +5ˆm 8 q ŝ 3ˆm 2 qŝ+9ˆm 4 qŝ 5ˆm 6 qŝ 15ŝ 2 18ˆm 2 qŝ 2 15ˆm 4 qŝ 2 +25ŝ 3 +25ˆm 2 qŝ 3 10ŝ 4) ˆλ ] ( 2 C eff ) C 10 2 û(ŝ, ˆm q ) [ 8 + 3û(ŝ, ˆm q) ( 2(1+ ˆm 2 q )(1 ˆm2 q )2 (1+14ˆm 2 q + ˆm4 q )ŝ (1+ ˆm2 q )ŝ2) (2 6ˆm2 q +4ˆm 4 q +4ˆm 6 q 6ˆm 8 q +2ˆm 10 q 2 5ŝ 12ˆm qŝ+34ˆm4 qŝ 12ˆm6 qŝ 5ˆm8 qŝ+3ŝ2 ˆλ +29ˆm 2 qŝ2 +29ˆm 4 qŝ2 +3ˆm 6 qŝ2 +ŝ 3 10ˆm 2 qŝ3 + ˆm 4 qŝ3 ŝ 4 ˆm 2 1 qŝ4 ) û(ŝ, ˆm q ) +4 ( 6+2ˆm 2 q +20ˆm4 q 12ˆm6 q 14ˆm8 q +10ˆm10 q +3ŝ+16ˆm 2 qŝ+62ˆm4 qŝ 56ˆm 6 qŝ 25ˆm8 qŝ+3ŝ2 +73ˆm 2 qŝ2 +101ˆm 4 qŝ2 +15ˆm 6 qŝ2 +5ŝ 3 26ˆm 2 qŝ3 + 5ˆm 4 qŝ 3 5ŝ 4 5ˆm 2 qŝ 4) ˆλ ] 2 C eff [ 8û(ŝ, ˆm q ) ( (1 ˆm 2 û(ŝ, ˆm q ) ŝ q) 2 (1+ ˆm 2 q )ŝ) +4(1 2ˆm 2 q + ˆm4 q ŝ ˆm2 qŝ) û(ŝ, ˆm q) ˆλ 1 +4 ( 5+30ˆm 4 q 40ˆm6 q +15ˆm8 q ŝ+21ˆm2 qŝ+25ˆm4 qŝ 45ˆm6 qŝ+13ŝ2 +22ˆm 2 qŝ2 +45ˆm 4 qŝ2 7ŝ 3 15ˆm ˆλ ] } 2 2 qŝ3) Re(C eff 9 )C eff 7. (9) û(ŝ, ˆm q ) We explain some aspect of the expression, (9): The branching ratio is normalized by B sl of decays B (X c,x u )lν l. We separate a combination of the KM factor V tq V tb 2 due to top-quark loop from the V cb 2 normalization factor B 0. The normalization constant B 0 is B 0 B sl 3α 2 16π 2 1 f(ˆm c )κ(ˆm c ), (10) where f(ˆm c ) is a phase space factor, and κ(ˆm c ) accounts for both the O(α s ) QCD correction to the semi-leptonic decay width and the leading order (1/m b ) 2 power correction, They are given in our Appendix explicitly. The Wilson coefficients depend on the top quark mass m t, the renormalization scale µ, and Λ QCD. Their dependences are studied in Tables 2, 3 and 4. As the value of µ (Λ QCD ) decreases (increases), C 7 and C 9 are getting larger, while C 10 is independent on µ. As m t increases, C 7, C 9 and C 10 all become 7

8 larger. The matrix elements also depend on the renormalization scale µ, and their dependence is partially cancelled by the given dependence of the Wilson coefficients. The overall dependence can be studied with the differential rate. In Figs. 1 and 2, we show the dependence of the differential rate on m t and µ. As the value of m t increases from 166 (GeV) to 184 (GeV), the differential rate increases about 20% at s 5 (GeV 2 ), as shown in Fig. 1. At around the same region of s, by changing µ from 2.5 (GeV) to 10 (GeV) the differential rate increases about 20%, as shown in Fig. 2. This observation is consistent with the result of Ref. [7], in which only the SD contribution has been analysed. The differential decay rate (9) is not a simple parton model result. It contains non-perturbative 1/m b 2 power corrections, which are denoted in Eq. (9) by the terms proportional to ˆλ 1 and ˆλ 2. The parameters ˆλ 1 and ˆλ 2 are related to the matrix elements of the higher derivative operators of heavy quark effective theory [5], [9]; B B h(id) h 2 B 2M B λ 1 = 2M B m b 2ˆλ1, 2 σµν G µν h B 6M B λ 2 = 6M B m b 2ˆλ2, (11) h i where B denotes the pseudoscalar B meson, D µ is the covariant derivative, and G µν is the QCD field strength tensor. V ts 3 The extraction of V td Presently we can constrain the value of (1 ρ) 2 +η 2 from B d B d mixing, while from B X u l ν we obtain some limit on ρ 2 + η 2. Using those given limits, we can show how the ratio 1 λ 2 dr ds depends on input KM values, (1 ρ)2 +η 2 and φ 1 (or β). As is well known, V tbv td are written with M d of B d B d mixing. It reads as Vtb V td = Aλ 3 (1 ρ) 2 +η 2 [ 6π 2 ] 1/2 1/2 M = G 2 F η QCD M2 W M d, (12) B B 1/2 f B Bd E(x t ) 1/2 d where E(x t ) is the Inami-Lim function [10] of the box diagram, and x t mt2 m W 2. It has the value E(x t ) = 2.58 for m t = 175 (GeV) and it varies from 2.38 to 2.78 as 8

9 m t varies from 166 (GeV) to 184 (GeV). We use the following values; η QCD = 0.55, m t = 175 (GeV), and the experimental constraints M d = 0.474±0.031(ps 1 ), V ub = λ ρ 2 +η 2 = 0.08±0.02. (13) V cb Then, thoseconstraints (13)aretranslatedinto thelimitsof(1 ρ) 2 +η 2 and ρ 2 +η 2 as (1 ρ) 2 +η 2 = 0.85(1±0.15) { 0.2/(B 1/2 B d f Bd (GeV)) } 2 = [0.59±0.09,1.3±0.20], ρ 2 +η 2 = 0.36±0.09. (14) Here we used B 1/2 B d f Bd = 0.2±0.04(GeV) to get the range of (1 ρ) 2 +η 2 [0.5,1.5]. We also used Aλ 2 = 0.041(±0.003), and λ = In Fig. 3, we show the present limit in the plane of (φ 1, (1 ρ) 2 +η 2 ). The horizontal axis corresponds to φ 1 (or β), and the unit is degree. The vertical axis corresponds to (1 ρ) 2 +η 2. The thin dashed line is obtained from the central values of V ub and the thin solid line is obtained from V td. The central value corresponds to (φ 1, (1 ρ) 2 +η 2 ) (20 o, 0.85). Now let us consider the ratio 1 dr(s). In the SU(3) limit, we expect that this λ 2 ds ratio approaches to the input value of (1 ρ) 2 +η 2 for the dileptonic invariant masssquared s far below the peak of charmonium resonances. If s is on the peak of resonances, the ratio becomes 1. In Fig. 4, we show the ratio 1 λ 2 dr(s) ds for two sets of the assumed input values of (φ 1, (1 ρ) 2 +η 2 ), one of whose (1 ρ) 2 + η 2 is 0.59, and the other is They correspond to their small (or large) values allowed from the present experimental result of B d B d mixing. The solid curve corresponds to (φ 1, (1 ρ) 2 +η 2 ) = (20 o, 0.59), and the dot-dashed curve corresponds to (20 o, 1.33). The ratios for the CP conjugate process B X q l + l are also shown, denoted by the long-dashed curve for (1 ρ) 2 +η 2 = 1.33, and by the dashed curve for (1 ρ) 2 +η 2 = They are obtained by reversing the sign of φ 1 in the corresponding B X q l + l process; i.e., φ 1 φ 1. They are labeled as ( 20 o, 1.33) and ( 20 o, 0.59) in Figure 4. To summarize the numerical results of Fig. 4 and Section 3: The predicted ratio at low invariant mass region s 1 (GeV 2 ) is very near to the our assumed input value of (1 ρ) 2 +η 2, while on the peak of the resonances 9

10 J/ψ, ψ, this ratio becomes almost 1, as we expected, i.e. 1 dr(s) λ 2 ds 1 (s m J/ψ 2,m ψ 2,..), (1 ρ) 2 +η 2 (s being away from m J/ψ 2,..). (15) In the intermediate region, there is a characteristic interference between the LD contribution and the SD contribution, which can be only derived from the detailed expression of the distributions, Eq. (9). The value of the ratio does not depend much on whether the decaying particles are B or B at any invariant mass-squared region. This ratio changes only a few % when we change the input parameters, m t and µ, within the range shown in Table 1. The dependences on m t and µ are almost cancelled away in the ratio. Therefore, the uncertainties in this ratio due to the input parameters are much smaller than the uncertainties in the differential decay rate itself, as shown in Tables 2, 3 and 4 as well as in Figures 1 and 2. In Fig. 4 the range between the dot-dashed curve and the solid curve corresponds to the value of the ratio 1 λ 2 dr(s) ds allowed from the present experimental result of B d B d mixing. Future experimental measurements on the ratio of the branching fractions, B(B X d l + l ) and B(B X s l + l ), can give much better alternative for determination of V td V ts without any hadronic uncertainties, limited only by experimental statistics. If 10 9 BB pairs are produced, the expected number of the events for B X d l + l in the range of 4m µ 2 < s < 6 GeV (m µ = muon mass) is about 100 for (1 ρ) 2 +η 2 = 0.59, and is about 220 for (1 ρ) 2 +η 2 = Therefore, the statistical accuracy of (1 ρ) 2 +η 2 determined from this method is about 7% 10% with the expected production of 10 9 BB pairs. We have assumed in our numerical analysis the flavor SU(3) symmetry with m d = m s. We estimated the corrections due to SU(3) breaking by varying m d from 0.01 GeV up to m s = 0.2 GeV. And we find the ratio dr decreases within ds % for the range 1 < s < 9 (GeV 2 ). Therefore, we conclude the SU(3) breaking effect does not affect the extraction of V td V ts at all. 10

11 Acknowledgements TM would like to thank G. Hiller, K. Ochi, T. Nasuno, and Y. Kiyo for correspondence and assistance of numerical computations. The preliminary version of this work was given [11] as a talk by TM at BCONF97, Hawaii, March The work of CSK was supported in part by the CTP, Seoul National University, in part by the BSRI Program, Ministry of Education, Project No. BSRI , and in part by the KOSEF-DFG large collaboration project, Project No The work of TM was partially supported by Monbusho International Scientific Research Program (No ), and that of AIS and TM was supported also by Grant-in-Aid for Scientific Research on Priorty Areas (Physics of CP violation) from the Ministry of Education and Culture of Japan. 11

12 References [1] M. Gronau and J.L. Rosner, Phys. Lett. B376 (1996) 205. [2] G. Buchalla and A.J. Buras, Phys. Rev. D54 (1996) [3] F. Krüger and L.M. Sehgal, Phys. Rev. D55 (1997) [4] C.S. Lim, T. Morozumi and A.I. Sanda, Phys. Lett. B218 (1989) 343; N.G. Deshpande, J. Trampetic and K. Panose, Phys. Rev. D39 (1989) 1461; P.J. O Donnell and H.K.K. Tung, Phys. Rev. D43 (1991) R2067; A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B273 (1991) 505; N. Paver and Riazuddin, Phys. Rev. D45 (1992) 978. [5] A. Falk, M. Luke, and M.J. Savage, Phys. Rev. D49 (1994) [6] A. Ali, G. Hiller, L. Handoko, and T. Morozumi, Phys. Rev. D55 (1997) [7] A.J. Buras and M. Münz, Phys. Rev. D52 (1995) 186, and references therein. [8] Z. Ligeti and M.B. Wise, Phys. Rev. D53 (1996) [9] I.I. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, Phys. Rev. Lett. 71 (1993) 496; A.V. Manohar and M.B. Wise, Phys. Rev. D49 (1994) 1310; B. Blok, L. Koyrakh, M. Shifman and A.I. Vainshtein, Phys. Rev. D49 (1994) 335; T. Mannel, Nucl. Phys. B423 (1994) 396; M. Neubert, Phys. Rev. D49 (1994) [10] T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297 [E. 65 (1981) 1772]. [11] C.S. Kim, T. Morozumi and A.I. Sanda, talk given at BCONF97, Hawaii, March 1997, hep-ph/ (June 1997). [12] M. Jeżabek and J. H. Kühn, Nucl. Phys. B320 (1989) 20. [13] C.S. Kim and A.D. Martin, Phys. Lett. B225 (1989) 186; A. Ali and E. Pietarinen, Nucl. Phys. B154 (1979) 519; N. Cabibbo, G. Corbò and L. Maiani, Nucl. Phys. B155 (1979)

13 Parameter Value m W (GeV) m Z (GeV) sin 2 θ W m s 0.2 (GeV) m d 0.01 (GeV) m c 1.4 (GeV) m b 4.8 (GeV) m t 175±9 (GeV) µ (GeV) Λ (5) QCD αqed (GeV) α s (m Z ) 0.117±0.005 B sl (10.4±0.4) % λ (GeV 2 ) λ (GeV 2 ) Table 1: Values of the input parameters used in the numerical calculations of the decay rates. Unless, otherwise specified, we use the central values. µ C 1 C 2 C 3 C 4 C 5 C 6 C eff 7 C NDR 9 C (0) Table 2: µ (in GeV) dependence of the Wilson coefficients used in the numerical calculations. The values of Λ QCD and m t are fixed at their central values. 13

14 Λ QCD 5 C 1 C 2 C 3 C 4 C 5 C 6 C eff 7 C9 NDR C (0) Table 3: Λ QCD 5 dependence of the Wilson coefficients used in the numerical calculations. The values of m t and µ are fixed at their central values. m t C eff 7 C9 NDR C Table 4: m t dependence of the Wilson coefficients used in the numerical calculations. The values of Λ QCD 5 and µ are fixed at their central values. Appendix A Functions g(z,ŝ), g(0,ŝ) and ω(ŝ) The functions g(z,ŝ) and g(0,ŝ) are given as g(z,ŝ) = 8 9 ln(m b µ ) 8 9 lnz y 2 9 (2+y) 1 y [ Θ(1 y)(ln 1+ ] 1 y 1 1 y iπ)+θ(y 1)2arctan 1, (16) y 1 with y = 4z 2 /ŝ, and g(0,ŝ) = ln(m b µ ) 4 9 lnŝ+ 4 iπ. (17) 9 The function ω(ŝ) represents the O(α s ) correction from the one-gluon exchange in the matrix element of O 9 [12]; ω(ŝ) = 2 9 π2 4 3 Li 2(ŝ) 2 3 lnŝln(1 ŝ) 5+4ŝ 3(1+2ŝ) ln(1 ŝ) 2ŝ(1+ŝ)(1 2ŝ) 5+9ŝ 6ŝ2 lnŝ+ 3(1 ŝ) 2 (1+2ŝ) 6(1 ŝ)(1+2ŝ). (18) 14

15 B Functions f(ˆm c ) and κ(ˆm c ) The phase space function for Γ(B X c lν) in the lowest order (i.e., parton model) f(ˆm c ) = 1 8 ˆm 2 c +8 ˆm 6 c ˆm 8 c 24 ˆm 4 c ln ˆm c. (19) And κ(ˆm c ) accounts for both the O(α s ) QCD correction to the semi-leptonic decay width and the leading order (1/m b ) 2 power correction, The function g(ˆm c ) is given [12],[13] as κ(ˆm c ) = 1 2α s(m b ) g(ˆm c )+ h(ˆm c) 3π 2m 2 b and finally the function h(ˆm c ) is given [9] as. (20) g(ˆm c ) = (π )(1 ˆm c) , (21) h(ˆm c ) = λ 1 + λ [ ˆm 2 f(ˆm c ) c 72ˆm 4 c +72ˆm6 c 15ˆm8 c 72ˆm4 c ln ˆm ] c, (22) where λ 1 (or µ 2 π ) and λ 2 (or µ 2 ) denote the matrix element of the higher derivative G operators of heavy quark effective theory, as defined in Eq. (11). 15

16 mt=175 mt=175 9 mt=175+9 db/ds * S ( GeV 2 ) Figure 1: Dilepton invariant mass spectrum, db ds (B X se + e ) The unit of s is (GeV 2 ). The solid line corresponds to m t = 175 (GeV). The short-dashed line corresponds to m t = The long-dashed line corresponds to m t =

17 15.0 (db/ds db 0 /ds)/db 0 /ds (%) µ=10 (GeV) µ=2.5 (GeV) S ( GeV 2 ) Figure 2: The % variation of dilepton invariant mass spectrum defined as ( db(µ) ds db 0 ds )/db 0 ; B ds 0 = B(µ = 5(GeV)). The solid line corresponds to µ = 10 and the dashed line corresponds to µ =

18 2.0 B d B d bar Mixing V ub 1.6 (1 ρ) 2 +η φ 1 (degree) Figure 3: The limit on (1 ρ) 2 +η 2 and φ 1 (or β). The horizontal axis corresponds to φ 1, and the unit is degree. The vertical axis corresponds to (1 ρ) 2 + η 2. The thin dashed line and the thin solid line are obtained from the central values of V ub and V td respectively. The thick solid lines are obtained from the allowed range of V td from B d Bd mixing. The thick dashed lines are obtained from the allowed range of V ub. 18

19 1.6 (20, 0.59) (20,1.33) ( 20,1.33) ( 20,0.59) 1.4 dr/ds/λ S ( GeV 2 ) Figure 4: The ratio 1 dr versus s λ 2 ds (GeV2 ) with two different input values of (1 ρ) 2 + η 2. The solid curve correspond to (φ 1, (1 ρ) 2 + η 2 ) = (20 o, 0.59), and the dot-dashed curve corresponds to (20 o, 1.33). The ratios for CP conjugate process B X q l + l are denoted by the dashed curve for (1 ρ) 2 + η 2 = 1.33, and by the long-dashed curve for (1 ρ) 2 +η 2 = They are obtained by reversing the sign of φ 1 in the corresponding B X q l + l process; i.e., φ 1 φ 1. 19